No. 6. 15. Marz 1909. 10. Jahrgang.
On the Present Status of the Radiation Problem
by A. Einstein
This Journal recently published contributions by Messrs.
H. A. Lorentz 1 , Jeans 2 , and Ritz 3 which help to simplify
the critical interpretation of the present status of this
extremely important problem. In the belief that it would
be useful if all those who have seriously thought about
this matter communicate their views, even if they have
not been able to arrive at a final result, I join in with the
1. The simplest form in which we can express the
currently understood laws of electrodynamics is the set of
Maxwell-Lorentz partial differential equations. I regard
the equations containing retarded functions, in contrast
to Mr. Ritz, 3 as merely auxiliary mathematical forms.
The reason I see myself compelled to take this view is
first of all that those forms do not subsume the energy
principle, while I believe that we should adhere to the
strict validity of the energy principle until we have found
important reasons for renouncing this guiding star. It
is certainly correct that Maxwell's equations for empty
space, taken by themselves, do not say anything, they
only represent an intermediary construct. But, as is well
known, exactly the same could be said about Newton's
equations of motion, as well as about any theory that
needs to be supplemented by other theories in order to
yield a picture of a complex of phenomena. What distin-
guishes the Maxwell-Lorentz differential equations from
the forms that contain retarded functions is the fact that
they yield an expression for the energy and the momen-
tum of the system under consideration for any instant
1 H. A. Lorentz, Phys. Zeit. 9, 562-563, 1908.
2 J. H. Jeans, Phys. Zeit. 9, 853-855, 1908.
3 W. Ritz, Phys. Zeit. 9, 903-907, 1908.
of time, relative to any unaccelerated coordinate system.
With a theory that operates with retarded forces it is not
possible to describe the instantaneous state of a system
at all without using earlier states of the system for this
description. For example, if a light source A had emitted
a light complex toward the screen B, but it has not yet
reached the screen B, then, according to theories operat-
ing with retarded forces, the light complex is represented
by nothing except the processes that have taken place in
the emitting body during the preceding emission. Energy
and momentum — if one does not want to renounce these
quantities altogether — must then be represented as time
Mr. Ritz certainly claims that experience forces us
to abandon these differential equations and introduce the
retarded potentials. However, his arguments do not seem
sound to me.
If one defines, in agreement with Ritz,
1 f ip(x',y',z',t- ^) , , ,
fi = — ^ c -±dx',dy',dz'
4ir J r
1 [ <p(x',y',z',t+l) , , ,
f 2 = — / — ^ ^-dx ,dy ,dz ,
47r J r
then both fi and / 2 are solutions of the equation
/3 = a\h + a 2 f2
Physikalische Zeitschrift. 10. Jahrgang. No. 6.
is also a solution if a\ + a 2 = 1. But it is not true that
the solution / 3 is a more general solution than f\ and
that one specializes the theory by setting a\ = 1, a 2 = 0.
f(x,y,z,t) = /1
amounts to calculating the electromagnetic effect at the
point x, y, z from those motions and configurations of the
electric quantities that took place prior to the time point
f(x,y,z,t) = f 2 ,
one determines the electromagnetic effects from the mo-
tions and configurations that take place after the time
In the first case the electric field is calculated from
the totality of the processes producing it, and in the sec-
ond case from the totality of the processes absorbing it.
If the whole process occurs in a (finite) space bounded
on all sides, then it can be represented in the form
as well as in the form
/ = h
f = h
If we consider a field that is emitted from the finite into
the infinite, we can, naturally, only use the form
f = h,
exactly because the totality of the absorbing processes
is not taken into consideration. But here we are dealing
with a misleading paradox of the infinite. Both kinds
of representation can always be used, regardless of how
distant the absorbing bodies are imagined to be. Thus,
one cannot conclude that the solution / = /1 is more
special than the solution ai/i + 02/2, where a\ + a 2 = 1.
That a body does not "receive energy from infinity
unless some other body loses a corresponding quantity
of energy" cannot be brought up as an argument either,
in my opinion. First of all, if we want to stick to experi-
ence, we cannot speak of infinity but only of spaces lying
outside the space considered. Furthermore, it is no more
permissible to infer irreversibility of the electromagnetic
elementary processes from the non-observability of such
a process than it is permissible to infer irreversibility of
the elementary processes of atomic motion from the sec-
ond law of thermodynamics.
2. Jeans' interpretation can he disputed on the
grounds that it might not be permissible to apply the
general results of statistical mechanics to cavities filled
with radiation. However, the law deduced by Jeans can
also be arrived at in the following way 1 .
According to Maxwell's theory, an ion capable of
oscillating about an equilibrium position in the direction
of the X-axis will, on average, emit and absorb equal
amounts of energy per unit time only if the following
relation holds between the mean oscillation energy E v
and the energy density of the radiation q v at the proper
frequency v of the oscillator:
where c denotes the speed of light. If the oscillating ion
can also interact with gas molecules (or, generally, with a
system that can be described by means of the molecular
theory) , then we must necessarily have, according to the
statistical theory of heat,
E u =
(R = gas constant, N = number of atoms in one gram-
atom, T = absolute temperature), if, on average, no en-
ergy is transferred by the oscillator from the gas to the
radiation space 2
From these two equations we arrive at
N C 3 "
i.e., exactly the same law that has also been found by
Messrs. Jeans and H. A. Lorentz 3
3. There can be no doubt, in my opinion, that
our current theoretical views necessarily lead to the law
1 Cf. A. Einstein, Ann. d. Phys. (4) 17, 133-136, 1905.
2 M. Planck, Ann. d. Phys. 1, 69, 1900. (ed: original publication has 99 instead of 69 as pg. ref. There is no Planck
paper with pg. ref. 99. The intended reference is M. Planck, Ann. d. Phys. 1, 69-122, 1900.) Vorlesungen iiber die
Theorie der Wdrmestrahlung. III. Kapitel. (ed: III. Kapitel refers to "Drifter Abschnitt.")
3 It should be explicitly noted that this equation is an irrefutable consequence of the statistical theory of heat. The
attempt, on p. 178 of the book by Planck just cited, to question the general validity of Equation II, is based — it seems
to me — only on a gap in Boltzmann's considerations, which has been filled in the meantime by Gibbs' investigations.
Physikalische Zeitschrift. 10. Jahrgang. No. 6.
propounded by Mr. Jeans. However, we can consider it as
almost equally well established that formula (III) is not
compatible with the facts. Why, after all, do solids emit
visible light only above a fixed, rather sharply defined
temperature? Why are ultraviolet rays not swarming ev-
erywhere if they are indeed constantly being produced at
ordinary temperatures? How is it possible to store highly
sensitive photographic plates in cassettes for a longtime if
they constantly produce short-wave rays? For further ar-
guments I refer to §166 of Planck's repeatedly cited work.
Thus, we must say that experience forces us to reject ei-
ther equation (I), required by electromagnetic theory, or
equation (II), required by statistical mechanics, or both
4. We must ask ourselves how Planck's radiation
theory relates to the theory which is indicated in 2. and
which is based on our currently accepted theoretical foun-
dations. In my opinion the answer to this question is
made harder by the fact that Planck's presentation of his
own theory suffers from a certain logical imperfection. I
will now try to explain this briefly.
a) If one adopts the standpoint that the irreversibil-
ity of the processes in nature is only apparent, and that
the irreversible process consists in a transition to a more
probable state, then one must first give a definition of
the probability W of a state. The only definition worthy
of consideration, in my opinion, would be the following:
Let A\, Ai . . . Ai be all the states that a closed sys-
tem at a certain energy content can assume, or, more
accurately, all the states that we can distinguish in such
a system with the help of certain auxiliary means. Ac-
cording to the classical theory, after a certain time the
system will assume one particular state (e.g., Ai) and
then remain in this state (thermodynamic equilibrium).
However, according to statistical theory the system will
keep assuming, in an irregular sequence, all these states
A\ . . . Ai 1 . If the system is observed over a very long time
period 0, there will be a certain portion t„ of this time
such that during t„ and during r„ only, the system occu-
pies the state A v . The quantity r„/0 will have a definite
limiting value, which we call the probability W that the
system has assumed state A v .
Proceeding from this definition, one can show that
the entropy must satisfy the equation
S = — log W + const,
where the constant is the same for all states of the same
b) Neither Mr. Boltzmann nor Mr. Planck have giv-
en a definition of W.
They set, purely formally, W = number of complex-
ions of the state under consideration.
If one now demands that these complexions be eq-
ually probable, where the probability of the complexion
is defined in the same way that we have defined the prob-
ability of the state under a) , one will obtain precisely the
definition for the probability of a state given under a);
however, the logically unnecessary element "complexion"
has been used in the definition.
Even though the indicated relation between S and
W is valid only if the probability of a complexion is de-
fined in the manner indicated, or in a manner equivalent
to it, neither Mr. Boltzmann nor Mr. Planck has defined
the probability of a complexion. But Mr. Boltzmann did
clearly realize that the molecular-theoretical picture he
had chosen dictated his choice of complexions in a quite
definite manner; he discussed this on pgs. 404 and 405 of
his paper "Uber die Beziehung..." 2 that appeared in the
Wiener Sitzungsberichten of 1877 3 . Similarly, Mr. Planck
would have had no freedom in the choice of complexions
in the resonator theory of radiation. He could have been
permitted to postulate the pair of equations
W = number of complexions
only if he had appended the condition that the complex-
ions must be chosen such that, in the statistically-based
theoretical model chosen by him, they had been found
to be equally probable. In this way he would have ar-
rived at the formula advocated by Jeans. Though every
physicist must rejoice that Mr. Planck disregarded these
requirements in such a fortunate manner, it should not
1 That only this last interpretation is tenable follows immediately from the properties of Brownian motion.
2 ed: This reference appears to be to the publication: L. Boltzmann, "Uber die Beziehung eines allgemeinen
mcchanischen Satzes zum zweiten Hauptsatz der Warmetheorie." Wien 1877, in: Sitzungberichte der mathematisch-
naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften, 76: 373-435. Reprinted in Boltzmann,
Wissenschftliche Abhandlungen, Leipzig, 1909, 3 vols, Vol. 2, pgs. 164-223.
3 Cf. also L. Boltzmann, Vorlesungen fiber Gastheorie (Lectures on Gas Theory), Vol. I, p. 40, lines 9-23.
Physikalische Zeitschrift. 10. Jahrgang. No. 6.
be forgotten that the Planck radiation formula is incom-
patible with the theoretical foundation from which Mr.
Planck started out.
5. It is simple to see how one could modify the
foundations of the Planck theory in order to have the
Planck radiation formula truly result from the theoretical
foundations. I will not present the pertinent derivations
here but will rather just refer to my papers on this sub-
ject 1 . The result is as follows: One arrives at the Planck
radiation formula if one
1. adheres to equation (I), given above, between res-
onator energy and radiation pressure, which Planck
derived from Maxwell's theory 2 ,
2. modifies the statistical theory of heat by the fol-
lowing assumption: A structure that is capable of
oscillations with the frequency v, and which, due
to its possession of an electric charge, is capable of
converting radiation energy into energy of matter
and vice verse, cannot assume oscillation states of
any arbitrary energy, but rather only such oscilla-
tion states whose energy is a multiple of hv. Where
h is the constant designated by Planck which ap-
pears in his radiation equation.
6. Since the modification of the foundations of
Planck's theory, just described, necessarily leads to very
profound changes in our physical theories, it is very im-
portant to search for the simplest possible mutually inde-
pendent interpretations of Planck's radiation formula as
well as of the radiation law in general, insofar as the lat-
ter may be assumed to be known. Two considerations on
this matter, which are distinguished by their simplicity,
shall be briefly described below.
Until now, the equation S = log W has been ap-
plied mainly to calculate the quantity W on the basis
of a more or less complete theory, and then to calcu-
late the entropy from W. However, this equation can also
be applied conversely, using empirically obtained entropy
values S v to obtain the statistical probability of the indi-
vidual states A v of an isolated system. A theory yielding
values for the probability of a state that differ from those
obtained in this way must obviously be rejected. An anal-
ysis of the kind indicated for determining certain statis-
tical properties of heat radiation enclosed in a cavity has
already been carried out by me in an earlier paper 3 , in
which I first presented the theory of light quanta. How-
ever, since at that time I started from Wien's radiation
formula, which is valid only in the limit (for large values
of ^), I shall present here a similar derivation which pro-
vides a simple interpretation of the meaning of Planck's
Let V and v be two interconnected spaces bounded
by diffuse completely reflecting walls. Radiation energy
within the frequency range dv is enclosed in these spaces.
H is the radiation energy existing instantaneously in V,
and j) the radiation energy existing instantaneously in v.
After some time the proportion ho : tjq = V : v will, to
within an approximation, be achieved. At an arbitrarily
chosen instant of time, r\ will deviate from rj 0 according
to a statistical law that is obtained directly from the
relation between S and W, then if one changes over to
dW = const x e« d-n.
Let £ and a denote the entropy of the radiation in
the two respective spaces and set n = r/ a + s, then we
dr\ = de
The last equation goes to, since
' d(S + a)
if one assumes that V is is very large compared with v,
2l di 2
S = const
If we content ourselves with the first non vanishing term
of the series, thus causing an error that is small, for larger
v, compared with the cube of the radiation wavelength,
dW = const x e
1»( d 2 <r \ _
From this we obtain for the mean value e 2 of the square
of the energy fluctuation of the radiation occurring in v,
1 A. Einstein, Ann. d. Phys. (4) 20, 1906 and Ann. d. Phys. (4) 22, 1907, §1.
2 This amounts to the same as assuming that the electromagnetic theory of radiation at least yields correct time
averages. This assumption can hardly be doubted in the light of the utility of this theory in optics.
3 Ann. d. Phys. (4) 17, 132-148, 1905.
Physikalische Zeitschrift. 10. Jahrgang. No. 6.
If the radiation formula is known, we can calculate a from
it 1 . Taking Planck's radiation formula as an expression
of experience, one obtains, after a simple calculation,
&-kv 2 dv
We have thus arrived at an easily interpretable expression
for the mean value of the fluctuations of the radiation
energy present in v. We shall now show that the current
theory of radiation is incompatible with this result.
According to the current theory, the fluctuations are
due solely to the circumstance that the infinitely many
rays traversing the space, which constitute the radiation
present in v, interfere with one another and thus provide
a momentary energy that is sometimes greater, some-
times smaller than the sum of the energies that the in-
dividual rays would provide if they were not interfering
with each other at all. We could thus exactly determine
the quantity e 2 by a consideration that is mathematically
somewhat complicated. We shall content ourselves here
with a simple dimensional consideration. The following
conditions must be satisfied:
1. The magnitude of the mean fluctuation depends only
on A (wavelength), dX, a and v, where a denotes the
radiation density associated with the wavelengths
(<7 dX = gdu).
2. Since the radiation energies of adjacent wavelength
ranges and volumes 2 are simply additive, and the
corresponding fluctuations are independent of each
other, at a given A and g, e 2 must be proportional
to the quantities dX and v.
3. e 2 has the dimension of energy squared.
The expression for e 2 is thereby completely deter-
mined up to a numerical factor (of order of magnitude
1). In this way one arrives at the expression a 2 X 4 vdX,
which upon introduction of the variables used above re-
duces to the second term of the formula for e 2 just de-
veloped. But we would have obtained solely this second
term for e 2 had we started out with the Jeans formula.
One would then also have to set equal to a constant of
order of magnitude 1, which corresponds to Planck's de-
termination of the elementary quantum 3 . Thus, the first
term of the above expression for e 2 , which for the visible
1 Cf. Planck's often cited book, Equation (230).
2 Naturally, only if these are large enough.
radiation surrounding us everywhere makes a far greater
contribution than the second one, is not compatible with
the current theory.
If one sets, following Planck, = 1, then the first
term, if present alone, would yield a fluctuation of the
radiation energy equal to that produced if the radiation
consisted of point quanta of energy hv moving indepen-
dently of each other. This can be shown by a simple cal-
culation. One should remember that the contribution of
the first term to the average percent fluctuation of energy
is the greater the smaller the energy 7y 0 , and that the
magnitude of this percent fluctuation yielded by the first
term is independent of the size of the space v over which
the radiation is distributed; I mention this in order to
show how fundamentally different the actual statistical
properties of radiation are from those to be expected on
the basis of our current theory, which is based on linear,
homogeneous differential equations.
7. In the foregoing we have calculated the fluctu-
ations of the energy distribution in order to obtain infor-
mation on the nature of thermal radiation. In what fol-
lows we shall briefly show how one can obtain analogous
results by calculating the fluctuations of the radiation
pressure, due to fluctuations of the momentum.
Let a cavity surrounded on all sides by matter of
absolute temperature T contain a mirror that can move
freely in the direction perpendicular to its normal . If we
imagine it to be moving with a certain velocity from the
outset, then, due to this motion, more radiation will be
reflected at its front than at it's back; hence, the radia-
tion pressure acting on the front will be greater than that
acting on the back. Thus, due to its motion relative to
the cavity radiation, the mirror will be acted upon by a
force comparable to friction, which little by little would
have to consume the momentum if there did not exist
a cause of motion exactly compensating the average for
the momentum lost through the above-mentioned fric-
tional force. To the irregular fluctuations of the energy
of a radiation space studied above, there also correspond
irregular fluctuations of the momentum, or irregular fluc-
tuations of the pressure forces exerted by the radiation
By carrying out the interference consideration indicated above, one obtains -M- = 1.
4 The motions of the mirror considered here are wholly analogous to the so-called Brownian motion of suspended
Physikalische Zeitschrift. 10. Jahrgang. No. 6.
on the mirror, which would have to set the mirror in
motion even if it had originally been at rest. The mean
speed of the mirror has then to be determined from the
entropy-probability relation, and the law of the above-
mentioned frictional forces from the radiation law, which
is assumed to be known. From these two results one then
calculates the effect of the pressure fluctuations, which
in turn makes it possible to draw conclusions concerning
the constitution of the radiation or — more precisely —
concerning the elementary processes of the reflection of
the radiation from the mirror.
Let v denote the velocity of the mirror at time t. Ow-
ing to the frictional force mentioned above, this velocity
decreases by in the small time interval r, where m
denotes the mass of the mirror and P the retarding force
corresponding to unit velocity of the mirror. Further, wc
denote by A the velocity change of the mirror during t
corresponding to the irregular fluctuations of the radia-
tion pressure. The velocity of the mirror at time t + t
v + A.
For the condition that, on average,
changed during r, we obtain
v shall remain un-
or, if we omit relatively infinitesimal quantities and take
into account that the average value of vA obviously van-
A 2 = — ^2.
In this equation v 2 can be replaced, using the equation
rnv 2 _ 1 RT
which can be derived from the entropy-probability equa-
tion. Before giving the value of the friction constant P,
we specialize the problem under consideration by assum-
ing that the mirror completely reflects the radiation of
a certain frequency range (between v and v + dv) and
is completely transparent to radiation of other frequen-
cies. By calculation omitted here for the sake of brevity,
one obtains from a purely electrodynamic investigation
the following equation, which is valid for any arbitrary
where g again denotes the radiation density at frequency
v, and / denotes the surface area of the mirror. By sub-
stituting the values obtained for v 2 and P, we get
If we transform this expression using Planck's radiation
formula, we obtain
The close connection between this relation and the
one derived in the last section for the energy fluctua-
tion (e 2 ) is immediately obvious 1 , and exactly analogous
considerations can be applied to it. Further, according
to the current theory, the expression must reduce to the
second term (fluctuation due to interference) . If the first
term alone were present, the fluctuations of the radiation
pressure could be completely explained by the assump-
tion that the radiation consists of independently moving,
not too extended, complexes of energy hv. In this case,
too, the formula says that in accordance with Planck's
formula the effects of the two mentioned causes of fluctu-
ation act like fluctuations (errors) arising from mutually
independent causes (additivity of the terms of which the
square of the fluctuation is composed).
8. In my opinion, the last two considerations con-
clusively show that the constitution of radiation must be
different from what we currently believe. It is true that,
as the excellent agreement of theory and experiment in
optics has proved, our current theory correctly yields the
time averages, which alone can be directly observed, but
it necessarily leads to laws on thermal properties of ra-
diation that prove to be incompatible with experience if
one maintains the entropy-probability relation. The dis-
crepancy between the phenomena and the theory is the
1 These relations can be written in the form (with -Sr = 1)
£ 2 = \ hgv +
8ttv 2 J
Physikalische Zeitschrift. 10. Jahrgang. No. 6.
more prominent the larger v and the smaller g. At small
q the temporal fluctuations of the radiation energy of a
given space or of the force of radiation pressure on a given
surface are much larger than expected from our current
We have seen that Planck's radiation law can be un-
derstood if one uses the assumption that the oscillation
energy of frequency v can occur only in quanta of magni-
tude hv. According to the aforesaid, it is not sufficient to
assume that radiation can only be emitted and absorbed
in quanta of this magnitude, i.e., that we are dealing with
a property of the emitting or absorbing matter only; con-
siderations 6. and 7. show that the fluctuations in the
spatial distribution of the radiation and in the radiation
pressure also occur as if the radiation consisted of quanta
of the indicated magnitude. Certainly, it cannot be as-
serted that the quantum theory follows from Planck's
radiation law as a consequence and that other interpre-
tations are excluded. However, one can well assert that
the quantum theory provides the simplest interpretation
of the Planck formula.
It should be emphasized that the considerations pre-
sented would in the main in no way lose their value if it
should turn out that Planck's formula is not valid; it is
precisely that part of Planck's formula which has been
adequately confirmed by experience (the Wien radiation
law valid in the limit for large ^) which leads to the
theory of the light quantum.
9. The experimental investigation of the conse-
quences of the theory of light quanta is, in my opinion,
one of the most important tasks that the experimental
physics of today must solve. The results obtained so far
can be divided into three groups
a) There are clues concerning the energy of those
elementary processes that are associated with the ab-
sorption or emission of radiation of a certain frequency
(Stokes' rule; velocity of cathode rays produced by light
or X-rays; cathode luminescence, etc). To this group also
belongs the interesting use Mr. Stark has made of the
theory of light quanta to elucidate the peculiar energy
distribution in the spectrum of a spectral line emitted
by channel rays 1 .
The method of deduction is always as follows: If one
elementary process produces another one, then the en-
ergy, of the latter is not larger than that of the former.
On the other hand, the energy of one of the two elemen-
tary processes is known (of magnitude hv) if the latter
consists in the absorption or emission of radiation of a
Especially interesting would be the study of excep-
tions to Stokes' law. In order to explain these exceptions,
one has to assume that a light quantum is emitted only
when the emission center in question has absorbed two
light quanta. The frequency of such an event, and thus
also the intensity of the emitted light having a smaller
wavelength than the producing one, will in this case have
to be proportional to the square of the intensity of the
exciting light at weak irradiation (according to the law
of mass action) , while according to Stokes' rule a propor-
tionality with the first power of the exciting light inten-
sity is to be expected at weak irradiation.
b) If the absorption 2 of each light quantum brings
about an elementary process of a certain kind, then ^ is
the number of these elementary processes if the quantity
of energy E of radiation of frequency v is absorbed.
Thus, for example, if the quantity E of radiation of
frequency v is absorbed by a gas being ionized, then it
is to be expected that gram molecules of the gas
will be ionized. This relation only appears to presume
the knowledge of A; for if Planck's radiation formula is
written in the form
Q = aV ~H !
e t — 1
then is the number of ionized gram molecules.
This relation, which I presented in my first paper 3
on this subject, has unfortunately remained unnoticed
c) The results noted in 5. lead to a modification of
the kinetic theory of specific heat 4 and to certain rela-
tions between the optical and the thermal behavior of
10. It seems difficult to set up a theoretical sys-
tem that interprets the light quanta in a complete fash-
ion, the way our current molecular mechanics in conjunc-
tion with the Maxwell-Lorentz theory is able to interpret
the radiation formula propounded by Mr. Jeans. That we
1 J. Stark, Phys. Zeit. 9, 767, 1908.
2 Of course, the analogous consideration holds also conversely for the production of light by elementary processes
(e.g., by collisions of ions).
3 Ann. d. Phys. (4) 17, 132-148, 1905, §9. ed: In this paper, (3 is defined as j3 = |.
4 A. Einstein, Ann. d. Phys. (4) 22, 1907, pgs. 180-190 and 800.
Physikalische Zeitschrift. 10. Jahrgang. No. 6.
are only dealing with a modification of our current the-
ory, not with its complete abolition, seems already to be
implied by the fact that Jeans' law seems to be valid
in the limit (for small ^). An indication as to how this
modification would have to be carried out is given by a
dimensional analysis carried out by Mr. Jeans a few years
ago, which is extremely important, in my opinion, and
which — modified in some points — I shall now recount in
Imagine that a closed space contains an ideal gas
and radiation and ions, and that owing to their charge,
the ions are able to mediate an energy exchange between
gas and radiation. In a theory of radiation linked with
the consideration of this system the following quantities
can be expected to play a role, i.e., to appear in the
expression to be obtained for the radiation density g :
a) the mean energy r\ of a molecular structure (up to
an unknown numerical factor like ^jf-),
b) the velocity of light c,
c) the elementary quantum e of electricity,
d) the frequency v.
From the dimension of g, by solely considering the
dimensions of the four quantities mentioned above, one
can determine in a simple way what the form of the ex-
pression for g must be. Substituting the value of ^ for
77, we obtain
Re 2 v
and where ip denotes a function that remains undeter-
mined. This equation contains the Wicn displacement
law, whose validity can hardly remain in doubt. This has
to be understood as a confirmation of the fact that apart
from the four quantities introduced above, no other di-
mensional quantities play a role in the radiation law.
From this we conclude that, except for dimension-
less numerical factors that appear in theoretical devel-
opments and of course cannot be determined by dimcn-
sional considerations, the coefficients ^ and appear-
ing in the equation for g must be numerically equal to the
coefficients appearing in the Planck (or Wien) radiation
formula. Since the above non determinable dimensionlcss
numerical factors are hardly likely to make a change in
1 The Planck formula reads: g = — .
the order of magnitude, we can set, within the order of
magnitude 1 is concerned
?l £ 1
It is the second of these equations which has been used
by Mr. Planck to determine the elementary quanta of
matter or electricity. Concerning the expression for h, it
should be noted that
h = 6 x 10
— = 7 x 10
There are three decimal places missing here. But, this
may be due to the fact that the dimensionless factors are
The most important aspect of this derivation is that
it relates the light quantum constant h to the elementary
quantum e of electricity. We should remember that the
elementary quantum e is a stranger in Maxwell-Lorcntz
electrodynamics 2 . Foreign forces must be enlisted in or-
der to construct the electron in the theory; usually, one
introduces a rigid framework to prevent the electron's
electrical masses from flying apart under the influence of
their electric interaction. The relation h = — seems to
me to indicate that the same modification of the theory
that will contain the elementary quantum e as a con-
sequence will also contain the quantum structure of ra-
diation as a consequence. The fundamental equation of
1 d 2 y
d 2 ip d 2 w
— — H — + — — ~
dx 2 dy 2 dz 2
d 2 <p
will have to be replaced by an equation in which the uni-
versal constant e (probably its square) also appears in a
coefficient. The equation sought (or the system of equa-
tions sought) must be homogeneous in its dimensions. It
must remain unchanged upon application of the Lorentz
transformation. It cannot be linear and homogeneous. It
must — at least if Jeans' law is really valid in the limit
Cf. Levi-Civita, Comptes Rendus, 1907, "Sur le mouvement etc."
Physikalische Zeitschrift. 10. Jahrgang. No. 6.
for small % — lead in the limit for large amplitudes to the
form D((p) = 0.
I have not yet succeeded in finding a system of equa-
tions fulfilling these conditions which would have looked
to me suitable for the construction of the elementary elec-
trical quantum and the light quanta. The variety of pos-
sibilities does not seem so great, however, for one to be
forced to shrink from this task.
From what has been said in this paper under 4. above,
the reader could easily get an incorrect impression about
the standpoint taken by Mr. Planck with regard to his
own theory of thermal radiation. I therefore deem it ap-
propriate to note the following.
In his book, Mr. Planck emphasized in several places
that his theory should not yet be viewed as something
complete and final. At the end of his introduction, for ex-
ample, he says verbatim: "I find it important, however, to
especially emphasize at this point the fact, as elaborated
in greater detail in the last paragraphs of the book, that
the theory developed here does not claim by any means
to be fully complete, even though, as I believe, it offers
a feasible approach by which to consider the processes
of energy radiation from the same viewpoint as those of
The pertinent discussions in my paper should not be
construed as an objection (in the strict sense of the word)
against Planck's theory, but rather as an attempt to for-
mulate and apply the entropy-probability principle more
rigorously than has been done untill now. A more rigor-
ous formulation of this principle was necessary because
without it the subsequent developments in the paper, in
which the molecular structure of radiation was deduced,
would not have been adequately substantiated. So that
my conception of the principle would not appear as cho-
sen somewhat ad hoc, or arbitrary, I had to show why
the existing formulation is not yet completely satisfying.
Bern, January 1909.
(Received on 23 January 1909.)